Prior for the time of nucleation: Specifying a prior density \(\rho (t)\)  is trickier, for the usual reason that an (improper) uniform prior for a continuous parameter is not reparametrization invariant.
However the symmetries of the problem at hand offer a natural solution. As argued by Jaynes [1]}, consistency requires that \(\rho (t)\)  should be invariant under all transformations
that leave our state of knowledge unchanged. Importantly, the master equation, from which our probabilities are derived, is invariant under time translations. A time translation can
justifiably have one of two effects on our state of knowledge:
